I'm not going to tell you (or your mother) what to do, but here are a few things to consider:
- It is now recommended to keep rats in groups of 3 at a minimum, because it allows them to have more harmonious and varied relationships. Of course, having 2 rats is still much better than having just 1, but only having 2 rats means that each rat is completely dependent on the other rat for social interaction. If the rats have very different personalities, that can lead to tension, fighting, or even just general unhappiness. Imagine being locked in a house for the rest of your life with only 1 other person for company, and you don't even get to choose who that other person is. Rats seem to do better in small groups rather than just pairs, so that they can form multiple social relationships and choose who they interact with.
- Having a group of 3 rats also means that when one of them dies — and rats can sicken and die very suddenly — the remaining 2 rats still have each other for company. When rats are kept in pairs, and one dies, the other rat can become extremely depressed. Some rats left on their own this way reportedly stop eating. Of course, not all rats react this badly, but it's impossible to know how badly a rat is going to react to loneliness until it's already happening. At that point, finding new friends for the lone rat becomes an emergency, and that is not a nice situation for anyone to be in. Starting with 3 rats means that you have more time to find suitable new friends to add to the group when one of them inevitably dies.
- Relatedly, you don't keep a
rat, and you don't even keep a
trio of rats. You end up keeping an ongoing group of rats, adding new rats to the group as old ones die, so that nobody is ever left alone.
- I've never had a hamster so I can't comment on whether or not rats smell more, but remember that rats are both bigger and usually live in larger groups than hamsters. More and bigger animals means more poo.
You can keep the smell down however: good cage layout, good choice of cage substrate, good diet, and cleaning the right amount helps a lot. (Obviously, too little cleaning means they smell, but too much cleaning means they panic about everything smelling weird and scent mark more to compensate.)
- Rats can cost a lot in vet bills. They are very prone to respiratory illness, tumours, and kidney and heart disease, just to name a few things. They are also quite good at choosing inconvenient times to suddenly become ill, which means you have to take them to an emergency vet, which costs even more. In the UK, a tumour removal can easily be over £100. The most common tumours occur in female rats, and can be significantly prevented by spaying (removing their uterus and ovaries), but that's usually a £100 operation as well. Male rats are less prone to tumours but still get them, and are more prone to hormonal aggression, which requires neutering, which is usually at least £50. Ongoing treatment for chronic respiratory illness, or kidney or heart disease, can easily get to over £200 over the course of a rat's life. For a trio of rats in the UK, I would personally recommend having £500 set aside, or at least available on a credit card, in case of expensive emergencies: they don't happen every day, but they happen often enough that I would not recommend having rats without that kind of financial safety net. You mentioned being dependent on your mother's approval in terms of smell and number of rats; if you're not financially independent, your mother also needs to know about and be prepared for the potential costs.